Worksheet for finding the greatest common factor (GCF) and least common multiple (LCM) of integer pairs.

To solve the problem of finding the Greatest Common Factor (GCF) and Least Common Multiple (LCM) for each pair of integers, we will follow these steps:

Step 1: Understand the Definitions

- Greatest Common Factor (GCF): The largest positive integer that divides both numbers without leaving a remainder.
- Least Common Multiple (LCM): The smallest positive integer that is a multiple of both numbers.

Step 2: Use Prime Factorization

For each pair of numbers:
1. Find the prime factorization of each number.
2. For GCF, take the product of the lowest powers of all common prime factors.
3. For LCM, take the product of the highest powers of all prime factors present in either number.

Step 3: Solve Each Pair

#### Pair 1: \(60\) and \(66\)
- Prime Factorization:
- \(60 = 2^2 \times 3 \times 5\)
- \(66 = 2 \times 3 \times 11\)
- GCF: The common prime factors are \(2\) and \(3\). The lowest powers are \(2^1\) and \(3^1\).
\[
\text{GCF} = 2 \times 3 = 6
\]
- LCM: The highest powers of all prime factors are \(2^2\), \(3^1\), \(5^1\), and \(11^1\).
\[
\text{LCM} = 2^2 \times 3 \times 5 \times 11 = 4 \times 3 \times 5 \times 11 = 660
\]

#### Pair 2: \(44\) and \(14\)
- Prime Factorization:
- \(44 = 2^2 \times 11\)
- \(14 = 2 \times 7\)
- GCF: The common prime factor is \(2\).
\[
\text{GCF} = 2
\]
- LCM: The highest powers of all prime factors are \(2^2\), \(7^1\), and \(11^1\).
\[
\text{LCM} = 2^2 \times 7 \times 11 = 4 \times 7 \times 11 = 308
\]

#### Pair 3: \(7\) and \(56\)
- Prime Factorization:
- \(7 = 7\)
- \(56 = 2^3 \times 7\)
- GCF: The common prime factor is \(7\).
\[
\text{GCF} = 7
\]
- LCM: The highest powers of all prime factors are \(2^3\) and \(7^1\).
\[
\text{LCM} = 2^3 \times 7 = 8 \times 7 = 56
\]

#### Pair 4: \(20\) and \(22\)
- Prime Factorization:
- \(20 = 2^2 \times 5\)
- \(22 = 2 \times 11\)
- GCF: The common prime factor is \(2\).
\[
\text{GCF} = 2
\]
- LCM: The highest powers of all prime factors are \(2^2\), \(5^1\), and \(11^1\).
\[
\text{LCM} = 2^2 \times 5 \times 11 = 4 \times 5 \times 11 = 220
\]

#### Pair 5: \(13\) and \(31\)
- Prime Factorization:
- \(13 = 13\)
- \(31 = 31\)
- GCF: There are no common prime factors.
\[
\text{GCF} = 1
\]
- LCM: The highest powers of all prime factors are \(13^1\) and \(31^1\).
\[
\text{LCM} = 13 \times 31 = 403
\]

#### Pair 6: \(8\) and \(42\)
- Prime Factorization:
- \(8 = 2^3\)
- \(42 = 2 \times 3 \times 7\)
- GCF: The common prime factor is \(2\).
\[
\text{GCF} = 2
\]
- LCM: The highest powers of all prime factors are \(2^3\), \(3^1\), and \(7^1\).
\[
\text{LCM} = 2^3 \times 3 \times 7 = 8 \times 3 \times 7 = 168
\]

#### Pair 7: \(16\) and \(60\)
- Prime Factorization:
- \(16 = 2^4\)
- \(60 = 2^2 \times 3 \times 5\)
- GCF: The common prime factor is \(2\). The lowest power is \(2^2\).
\[
\text{GCF} = 2^2 = 4
\]
- LCM: The highest powers of all prime factors are \(2^4\), \(3^1\), and \(5^1\).
\[
\text{LCM} = 2^4 \times 3 \times 5 = 16 \times 3 \times 5 = 240
\]

Final Answer

\[
\boxed{
\begin{array}{|c|c|c|}
\hline
& \text{GCF} & \text{LCM} \\
\hline
1. 60, 66 & 6 & 660 \\
\hline
2. 44, 14 & 2 & 308 \\
\hline
3. 7, 56 & 7 & 56 \\
\hline
4. 20, 22 & 2 & 220 \\
\hline
5. 13, 31 & 1 & 403 \\
\hline
6. 8, 42 & 2 & 168 \\
\hline
7. 16, 60 & 4 & 240 \\
\hline
\end{array}
}
\]